I'm not a linguist, so please forgive any apparent presumption on my part in venturing an answer here. I'm confident about what I'm saying but more interested in the discussion of this answer with experts, than I am in finding agreement.
Conversion seems to apply only to morphological changes to a word's base form through affixation, or zero-derivation where the base form is just used as a different PoS, for example run in *I run," (verb), and I had a good run, (noun). But the PoS designation can only be made once a word is functioning in a sentence. Otherwise, it is not possible to say that a word like run, alone in its base form, has any part of speech, unless it only functions as a single part of speech, like the noun car or the verb teach, which have very narrow functional scope - car is always a noun and teach is always a verb. There cannot be any conversion when a word such as run is used as a noun, it is simply a word that con function as either a noun or a verb.
The -ing participle is a morphological form of its base word which is always a verb. The -ing participle has not undergone conversion unless it is used as a different part of speech from its base, in other words, when it is used as an adjective or noun in a noun phrase. Otherwise, it can be used in the progressive aspect of finite verb phrases or in non-finite verb phrases, or used as a bare participle as a subject or object. In the latter case as a bare participle it derives it's thingness from being used as a subject or object but it is still simply a verb form.
I just don't see any reason to say there is conversion when bare -ing participles are used as subjects or objects. If one were to assert that conversion does indeed happen at the sentence phrase level, then we would have to say that clauses used as subjects or objects also undergo conversion and become nouns. I know that some folks do say this, e.g. the noun clause (aka content clause). In all of these cases (bare participles and relative clauses as subjects or objects) calling them anything but verb forms or clauses, is designating these forms as a PoS (noun) which they are not. They are verbs or clauses as subjects or objects. To do so confuses sentence level roles and their S|V|C relations with PoS designations. Subjects and objects are always things but they are not always nouns, and just because a verb from or clause can take these sentence level roles does not render them nouns.
I think the only true example of conversion is in sentence number 2.
I will be a contestant in the baking competition. (as an adjective in
a compound noun phrase)
Because the participle form is used in the adjective position in a noun phrase it converts from participle to adjective and in doing so it loses its ability behave like a verb - it cannot be modified as a verb any more - no adverbial or object relations. I don't think present participle adjectives can even take a qualifying adverb as a modifier, as many base adjectives can.
Conversion (nominalization) would also happen for present participles used in the noun position in a noun phrase:
some difficult programming
a very old building,
All of these verb forms are nominalized by the constraint of the noun phrase structure they are used in and they lose their ability to function as verbs. But -ing participles used subjects and objects without such noun phrase constraints can still take adverbial complements and direct objects in ways that participles used in noun phrases cannot.
Otherwise, participles used alone as a subject or object derive their thingness from the role they play as subject or object, not through conversion. They can only be identified as a verb phrase because they are verb forms and as word phrases have none of the structural elements of a noun phrase; no determiners, adjectives, and no nouns. The only thing they have in common with nouns is their thingness, which is inherited from their syntactical role as subject or object, but in terms of word class or PoS, they can only be identified as verbs.
Subjects and objects are always things but they are roles in sentences that can be played by different types of word phrases; noun phrases, verb phrases (participles and infinitives) and even clauses can function as subjects and objects.